We will prove that both the partial sums with odd number of terms, and with even number of terms, converge to the same number L. Thus the usual partial sum also converges to L.

The odd partial sums decrease monotonically:

while the even partial sums increase monotonically:

both because an decrease monotonically with n.

Moreover, since an are positive, . Thus we can collect these facts to form the following suggestive inequality:

Now, note that a1 − a2 is a lower bound of the monotonically decreasing sequence S2m+1, the monotone convergence theorem then implies that this sequence converges as m approaches infinity. Similarly, the sequence of even partial sum converges too.

for any m. This means the partial sums of an alternating series also "alternates" above and below the final limit. More precisely, when there are odd (even) number of terms, i.e. the last term is a plus (minus) term, then the partial sum is above (below) the final limit.

This understanding leads immediately to an error bound of partial sums, shown below.